Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}4x+2y &= -8 \\ -6x+y &= -3\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}-4x-2y &= 8\\ -12x+2y &= -6\end{align*}$ Add the top and bottom equations. $-16x = 2$ Divide both sides by $-16$ and reduce as necessary. $x = -\dfrac{1}{8}$ Substitute $-\dfrac{1}{8}$ for $x$ in the top equation. $4( -\dfrac{1}{8})+2y = -8$ $-\dfrac{1}{2}+2y = -8$ $2y = -\dfrac{15}{2}$ $y = -\dfrac{15}{4}$ The solution is $\enspace x = -\dfrac{1}{8}, \enspace y = -\dfrac{15}{4}$.